Regularity results for nonlinear Young equations and applications

In this paper we provide sufficient conditions which ensure that the nonlinear equation dy(t)=Ay(t)dt+σ (y(t))dx(t) , t∈ (0,T] , with y(0)=ψ and A being an unbounded operator, admits a unique mild solution such that y(t)∈ D(A) for any t∈ (0,T] , and we compute the blow-up rate of the norm of y ( t ) as t→ 0^+ . We stress that the regularity of y is independent of the smoothness of the initial datum ψ , which in general does not belong to D ( A ). As a consequence we get an integral representation of the mild solution y which allows us to prove a chain rule formula for smooth functions of y .